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LoadPackage( "GradedRingForHomalg", false );
R := GradedRing( HomalgFieldOfRationalsInDefaultCAS( ) * "a,b,c" );
LoadPackage( "GradedModules", false );
O := n -> (R * 1)^n;
## p[1] := (0:0:1), p[2] := (0:1:0), p[3] := (1:0:0)
p := [ "[ a, b ]", "[ a, c ]", "[ b, c ]" ];
## are s distinct points in P2
s := Length( p );
## with defining ideals
p := List( p, q -> GradedRightSubmodule( q, R ) );
## and multiplicities
r := [ 2, 2, 2 ];
curve := IntersectWithMultiplicity( p, r );
Curve := MatrixOfGenerators( curve );
## a random plane curve of degree d with s ordinary singularities and genus g
d := 6;
g := Binomial( d - 1, 2 ) - Sum( [ 1 .. s ], i -> Binomial( r[i], 2 ) );
F := Curve * RandomMatrix( O( -d ), curve );
## adjunction: L( d - 3; (r[1]-1) * p[1], ..., (r[s]-1) * p[s] );
can := IntersectWithMultiplicity( p, r - 1 );
can := SubmoduleGeneratedByHomogeneousPart( d - 3, can );
## S: Proj( S ) = P^{g-1}
S := CoefficientsRing( R ) * [ "x", [ 0 .. g - 1 ] ];
images := EntriesOfHomalgMatrix( MatrixOfGenerators( can ) );
T := R / EntriesOfHomalgMatrix( F );
f := RingMap( images, S, T );
SetDegreeOfMorphism( f, 0 );
IC := KernelSubobject( f );
OC := 1 * S / IC;
betti := BettiTable( Resolution( Int( g / 2 ) - 1, OC ) );
## a tacnode leads to the same betti diagram
line := HomalgMatrix( "[ a - b ]", 1, 1, R );
line := Subobject( line, ( R * 1 )^0 );
can3 := Intersect( line + p[1]^2, p[3] );
can3 := SubmoduleGeneratedByHomogeneousPart( 3, can3 );
## A: the Koszul dual ring
A := KoszulDualRing( S, [ "e", [ 0 .. g - 1 ] ] );
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